The Berezinskii–Kosterlitz–Thouless transition (BKT transition) is a phase transition in the two-dimensional (2-D) XY model. It is a transition from bound vortex-antivortex pairs at low temperatures to unpaired vortices and anti-vortices at some critical temperature. The transition is named for condensed matter physicists Vadim Berezinskii, John M. Kosterlitz and David J. Thouless. BKT transitions can be found in several 2-D systems in condensed matter physics that are approximated by the XY model, including Josephson junction arrays and thin disordered superconducting granular films. More recently, the term has been applied by the 2-D superconductor insulator transition community to the pinning of Cooper pairs in the insulating regime, due to similarities with the original vortex BKT transition.
Contents
- XY model
- KT transition disordered phases with different correlations
- Role of vortices
- Informal description
- Almost rigorous analysis
- Books
- References
Work on the transition led to the 2016 Nobel Prize in Physics being awarded to Thouless, Kosterlitz and Duncan Haldane.
XY model
The XY model is a two-dimensional vector spin model that possesses U(1) or circular symmetry. This system is not expected to possess a normal second-order phase transition. This is because the expected ordered phase of the system is destroyed by transverse fluctuations, i.e. the Nambu-Goldstone modes (see Goldstone boson) associated with this broken continuous symmetry, which logarithmically diverge with system size. This is a specific case of what is called the Mermin–Wagner theorem in spin systems.
Rigorously the transition is not completely understood, but the existence of two phases was proved by McBryan & Spencer (1977) and Fröhlich & Spencer (1981).
KT transition: disordered phases with different correlations
In the XY model in two dimensions, a second-order phase transition is not seen. However, one finds a low-temperature quasi-ordered phase with a correlation function (see statistical mechanics) that decreases with the distance like a power, which depends on the temperature. The transition from the high-temperature disordered phase with the exponential correlation to this low-temperature quasi-ordered phase is a Kosterlitz–Thouless transition. It is a phase transition of infinite order.
Role of vortices
In the 2-D XY model, vortices are topologically stable configurations. It is found that the high-temperature disordered phase with exponential correlation is a result of the formation of vortices. Vortex generation becomes thermodynamically favorable at the critical temperature
Many systems with KT transitions involve the dissociation of bound anti-parallel vortex pairs, called vortex–antivortex pairs, into unbound vortices rather than vortex generation. In these systems, thermal generation of vortices produces an even number of vortices of opposite sign. Bound vortex–antivortex pairs have lower energies than free vortices, but have lower entropy as well. In order to minimize free energy,
Informal description
There is an elegant thermodynamic argument for the KT transition. The energy of a single vortex is
When
The KT transition can be observed experimentally in systems like 2D Josephson junction arrays by taking current and voltage (I-V) measurements. Above
Almost rigorous analysis
The following discussion uses field theoretic methods, and is very nearly rigorous (see the "~" formulation below). Assume a field φ(x) defined in the plane which takes on values in
The energy is given by
and the Boltzmann factor is exp(−βE).
If we take the contour integral
Now,
Unless
When
This is exactly the energy function for a two-dimensional Coulomb gas; the scale L contributes nothing but a constant.
Assume the case with only one vortex of multiplicity one and one vortex of multiplicity −1. At low temperatures, i.e. large β, because of the Boltzmann factor, the vortex–antivortex pair tends to be extremely close to one another. In fact, their separation would be around the cutoff scale. With more vortex–antivortex pairs, we have a collection of vortex-antivortex dipoles. At large temperatures, i.e. small β, the probability distribution swings the other way around and we have a plasma of vortices and antivortices. The phase transition between the two is the Kosterlitz–Thouless phase transition.